The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra

1928 ◽  
Vol 24 (3) ◽  
pp. 426-437 ◽  
Author(s):  
D. R. Hartree
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Coulomb Field ◽  
Practical Method ◽  
Optical Spectra ◽  
Central Field ◽  
Wave Mechanics ◽  
In Series ◽  
Characteristic Values ◽  
Image Position

In two recent papers the writer has given an account of a practical method of finding the characteristic values and functions of Schrödinger's wave equations for a given non-Coulomb central field. For terms of optical spectra the method is effectively the following. We take the wave equation in the formand require the values of ɛ for the solutions which are zero at the origin and at r = ∞. We consider the result of integrating this equation outwards from P = 0 at r = 0 to a radius r0 at which the deviation from a Coulomb field is negligible, and inwards from P = 0 at r = ∞ to the same radius, with a given value of ɛ; the characteristic values are those values for which these two solutions join smoothly on to one another, i.e. for which they have the same value of η = −P′/P at this radius. For a given ɛ, the solution zero at the origin depends on the particular atom; the solution zero at infinity can be expressed in a form independent of any particular atom.

scholarly journals The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

1928 ◽  
Vol 24 (1) ◽  
pp. 89-110 ◽  
Author(s):  
D. R. Hartree
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Initial Conditions ◽  
Energy Parameter ◽  
Coulomb Field ◽  
Characteristic Functions ◽  
Central Field ◽  
Wave Mechanics ◽  
Numerical Work ◽  
Characteristic Values

The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).The method used for integrating the equations numerically is outlined (§ 9).

The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part IV. Further Results relating to Terms of the Optical Spectrum

1929 ◽  
Vol 25 (3) ◽  
pp. 310-314 ◽  
Author(s):  
D. R. Hartree
Keyword(s):  
Wave Equation ◽  
Optical Spectrum ◽  
Coulomb Field ◽  
Central Field ◽  
Wave Mechanics

The purpose of this note is to extend in two respects some previous work of the writer on the wave equation of a particle in a central field of force which is a Coulomb field for sufficiently large r.

The Dispersion Electrons in the One-Electron Problem

1929 ◽  
Vol 25 (3) ◽  
pp. 323-330
Author(s):  
J. Hargreaves
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Dipole Moment ◽  
Optical Spectrum ◽  
Wave Equations ◽  
Simple Wave ◽  
Incoherent Scattering ◽  
Central Field ◽  
Dispersion Formula ◽  
The One

In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.

scholarly journals Lifespan of solutions to wave equations in de Sitter spacetime

2018 ◽  
Vol 148 (6) ◽  
pp. 1313-1330 ◽  
Author(s):  
Weiping Yan
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave Equation ◽  
Finite Time ◽  
Wave Equations ◽  
Blow Up ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Sitter Spacetime ◽  
Image Position ◽  
Lifespan Estimate

We consider the finite-time blow-up of solutions for the following two kinds of nonlinear wave equation in de Sitter spacetime:This proof is based on a new blow-up criterion, which generalizes that by Sideris. Furthermore, we give the lifespan estimate of solutions for the problems.

scholarly journals Interaction of mesons with a potential field

1948 ◽  
Vol 193 (1035) ◽  
pp. 559-579 ◽  
Keyword(s):  
Exact Solution ◽  
Wave Equation ◽  
Elastic Scattering ◽  
Electrostatic Field ◽  
Potential Field ◽  
Born Approximation ◽  
Wave Equations ◽  
Scattering Problem ◽  
Coulomb Field ◽  
Radial Wave

The Takata-Saketani wave equation for mesons is developed to allow the solution of a variety of problems concerning the behaviour of charged vector mesons in an electrostatic field. In particular the form of the radial wave equations is discussed. The general elastic scattering problem is also formulated, and the Born approximation for the phases is discussed for the Coulomb field, for which, as is known, no exact solution is possible. The exact treatment of scattering by a spherical well is considered. This is the only case for which exact treatment in terms of known functions seems practicable, and might be of service in indicating the limits of the validity of approximate processes in other cases.

On the numerical solution of a class of partial differential equations

1958 ◽  
Vol 54 (2) ◽  
pp. 214-218 ◽  
Author(s):  
A. S. Douglas
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Ground State ◽  
Wave Equations ◽  
Ground State Solution ◽  
Electronic Computing ◽  
Image Position

ABSTRACTFor a suitable choice of E*, the solution as t becomes large of the equationwhere Y is given independent of t over the space boundaries, tends to the ground state solution of the wave equationwith the same boundary conditions on P as on Y. As a preliminary to using this relation to solve wave equations in more than one variable, the solution of the equationhas been studied. Methods of numerical solution are discussed, and the convergence of these is examined. Some practical experiments using an electronic computing machine are described.

Some Calculations relevant to the Quantum Defect in the Extended Ritz Formula

1929 ◽  
Vol 25 (3) ◽  
pp. 315-322 ◽  
Author(s):  
J. Hargreaves
Keyword(s):  
Wave Equation ◽  
Coulomb Field ◽  
Quantum Defect ◽  
Image Position

The object of this paper is to make certain calculations which will be required in the estimation of the quantum defect. Hartree has defined certain solutions Gi(σ, ζ), Hi(σ, ζ) of the wave equation for an electron in a Coulomb field, viz.

The role of differential phase contrast in electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 176-177
Author(s):  
E.M. Waddell ◽  
J.N. Chapman ◽  
R.P. Ferrier
Keyword(s):  
Phase Contrast ◽  
Practical Method ◽  
Position Vector ◽  
Differential Phase ◽  
Pure Phase ◽  
Differential Phase Contrast ◽  
The Difference ◽  
Image Position ◽  
First Moment

Dekkers and de Lang (1977) have discussed a practical method of realising differential phase contrast in a STEM. The method involves taking the difference signal from two semi-circular detectors placed symmetrically about the optic axis and subtending the same angle (2α) at the specimen as that of the cone of illumination. Such a system, or an obvious generalisation of it, namely a quadrant detector, has the characteristic of responding to the gradient of the phase of the specimen transmittance. In this paper we shall compare the performance of this type of system with that of a first moment detector (Waddell et al.1977).For a first moment detector the response function R(k) is of the form R(k) = ck where c is a constant, k is a position vector in the detector plane and the vector nature of R(k)indicates that two signals are produced. This type of system would produce an image signal given bywhere the specimen transmittance is given by a (r) exp (iϕ (r), r is a position vector in object space, ro the position of the probe, ⊛ represents a convolution integral and it has been assumed that we have a coherent probe, with a complex disturbance of the form b(r-ro) exp (iζ (r-ro)). Thus the image signal for a pure phase object imaged in a STEM using a first moment detector is b2 ⊛ ▽ø. Note that this puts no restrictions on the magnitude of the variation of the phase function, but does assume an infinite detector.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

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